Write the equation for a parabola with a focus at $(-4,8)$ and a directrix at $x=-6$. $x=$
Answer: The strategy A parabola is defined as the set of all points that are the same distance away from a point (the focus) and a line (the directrix). Let $(x,y)$ be a point on the parabola. Then the distance between $(x,y)$ and the focus, $(-4,8)$, is equal to the distance between $(x,y)$ and the directrix, $x=-6$. Once we find these distances, we can equate them in order to derive the equation of our parabola. Finding the distances from $(x,y)$ to the focus and the directrix The distance between $(x,y)$ and $(-4,8)$ is $\sqrt{(x+4)^2+(y-8)^2}$. [How did we find that?] Similarly, the distance between $(x,y)$ and the line $x=-6$ is $\sqrt{(x+6)^2}$. [How did we know that?] Deriving the formula by equating the distances $\begin{aligned} \sqrt{(x+6)^2} &= \sqrt{(x+4)^2+(y-8)^2} \\\\ (x+6)^2 &= (x+4)^2+(y-8)^2 \\\\ {x^2}+12x{+36} &= {x^2}{+8x}+16+(y-8)^2\\\\ 12x{-8x}&=(y-8)^2+16{-36} \\\\ 4x&=(y-8)^2-20\\\\ x&=\dfrac{(y-8)^2}{4}-5\end{aligned}$ The answer The equation of our parabola is $x=\dfrac{(y-8)^2}{4}-5$. Here is the graph of our parabola. As expected, the distance between a point on the parabola, $(x,y)$, and the focus is the same as the distance between $(x,y)$ and the directrix. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${10}$ ${11}$ ${12}$ ${13}$ ${14}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ $y$ $x$ ${(x,y)}$